def count_solutions(a, b, c, n):
    count = 0
    for x in range(n // a + 1):
        for y in range((n - a * x) // b + 1):
            z = (n - a * x - b * y) // c
            if a * x + b * y + c * z == n:
                count += 1
    return count
a = 2
b = 3
c = 5
n = 10
print(count_solutions(a, b, c, n))

from math import gcd


def count_solutions(a, b, c, n):
    if n % gcd(a, b, c) == 0:
        return float('inf')
    else:
        return 0

def get_solutions(coefficients, total):
    partial_solutions: list[tuple[int, tuple[int, ...], int]] = [(0, (), total)]

    while partial_solutions:
        amplitude, variables, total = heappop(partial_solutions)

        if len(variables) == len(coefficients):
            if total == 0:
                yield variables
        else:
            unset_coefficients = coefficients[len(variables):]
            if total % gcd(0, *unset_coefficients) == 0:
                if len(unset_coefficients) == 1:
                    yield variables + (total // unset_coefficients[-1],)
                else:
                    heappush(partial_solutions, (amplitude, variables + (0,), total))

        if len(variables) > 0:
            *variables, variable = variables
            variables = tuple(variables)
            coefficient = coefficients[len(variables)]
            if variable >= 0:
                variables_ = variables + (variable + 1,)
                heappush(partial_solutions, (max(amplitude, variable + 1),
                                             variables_,
                                             total - coefficient))
            if variable <= 0:
                variables_ = variables + (variable - 1,)
                heappush(partial_solutions, (max(amplitude, -(variable - 1)),
                                             variables_,
                                             total + coefficient))

def get_natural_solutions(coefficients, total):
    partial_solutions: list[tuple[tuple[int, ...], int]] = [((), total)]

    while partial_solutions:
        variables, total = partial_solutions.pop()

        if len(variables) == len(coefficients):
            if total == 0:
                yield variables
            continue

        unset_coefficients = coefficients[len(variables):]
        if total % gcd(0, *unset_coefficients) == 0:
            variable = total // unset_coefficients[0]
            partial_solutions.append((variables + (variable,), total - variable * unset_coefficients[0]))

        if len(variables) > 0 and variables[-1] > 0:
            *variables, variable = variables
            variables = tuple(variables)
            coefficient = coefficients[len(variables)]
            partial_solutions.append((variables + (variable - 1,), total + coefficient))


def count_natural_solutions(coefficients, total):
    return sum(1 for _ in get_natural_solutions(coefficients, total))


def main():
    for variables in get_natural_solutions((7, 11, 13, 3), 47):
        print(variables)

def get_natural_solutions(coefficients, total):
    if len(coefficients) == 0:
        if total == 0:
            yield ()
        return

    if total % gcd(0, *coefficients) != 0:
        return

    for variable in range(0, total // coefficients[-1] + 1):
        for variables in get_natural_solutions(coefficients[:-1], total - variable * coefficients[-1]):
            yield variables + (variable, )